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Transforming Upper-Division Quantum Mechanics:
Learning Goals and Assessment
Steve Goldhaber, Steven Pollock, Mike Dubson, Paul Beale and Katherine Perkins
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
Abstract. In order to help students overcome documented difficulties learning quantum mechanics (QM) concepts, we have
transformed our upper-division QM I course using principles of learning theory and active engagement. Key components of
this process include establishing learning goals and developing a valid, reliable conceptual assessment tool to measure the
extent to which students achieve these learning goals. The course learning goals were developed with broad faculty input, and
serve as the basis for the design of the course assessment tool. The development of the assessment tool has included significant
faculty input and feedback, twenty-one student interviews, a review of PER literature, and administration of the survey to two
semesters of QM I students as well as to a cohort of graduate students. Here, we discuss this ongoing development process
and present initial findings from our QM class for the past two semesters.
Keywords: physics education research, quantum mechanics, assessment
PACS: 01.30.la,01.40.Fk,01.30.Cc,01.40.Di
INTRODUCTION
For the last decade, there has been a new focus in physics
education research aimed at undergraduate quantum mechanics (QM).[1] In recent years a growing body of evidence has shown that students experience a number of
difficulties learning upper-division QM[2, 3]. In order to
address these difficulties, the physics department at CUBoulder, using findings from lower-division PER[4] as
well as early results from other reform efforts[3], initiated an effort to reform the QM I course. Normally taken
by 35 - 50 junior physics and engineering physics majors,
this course is the first part of a two-semester sequence.
Topics include the Schrödinger equation, 1-D solutions,
operator methods, angular momentum and the hydrogen
atom. Classroom instruction using the new methods began in the Spring 2008 semester. Since then, the following changes have been made to the class structure:
The authors and colleagues have developed over
250 concept test questions and other activities to
promote student engagement during lectures.[8]
• The course has adopted an optional weekly recitation session where students work through a tutorial in small groups. Each hour-long session is facilitated by the course instructor and a learning
assistant.[5] A few of the tutorials were developed
by the authors with the majority coming from the
University of Washington.[3, 6]
• Homework problems were augmented to include
more justification of reasoning, estimation, and
tasks requiring students to make connections
between mathematical representations, physical meaning and other representations (e.g.,
graphical).[7]
•
•
Instructor office-hours were replaced with a cooperative, weekly session where students worked in
groups on the homework.
In addition to the changes outlined above, two other
initiatives helped to focus the reform efforts as well as
to track student achievements and learning difficulties:
course learning goals and an assessment tool.
LEARNING GOALS
In the Summer of 2008, we began the process of creating learning goals for our QM I course by interviewing
faculty who had experience teaching this course as well
as a few faculty who had taught the preceding and following courses in the sequence. These interviews were
followed by a series of meetings with interested faculty
members who discussed learning goals both for specific
course content and for general skills important to faculty.
Overall, eighteen faculty members contributed their time
and expertise to this process. An example of a contentspecific goal is “Given a wave function and an observable
operator, students will be able to calculate that operator’s
expectation value” while an example of a general skilloriented goal is “Students should be able to sketch the
physical parameters of a problem (e.g., wave function,
potential, probability distribution), as appropriate for a
particular problem.”[8]
While the learning goals developed during the process
described above cover much of what is normally taught
in an upper-division quantum mechanics course, it became apparent, both in the interviews and in the group
meetings that there are certain areas of both the pedagogy and the interpretation of quantum mechanics where
physics faculty have widely divergent views. This situation complicates efforts to reform instruction in quantum
mechanics and is discussed in another paper.[9]
Formal classroom assessments (i.e., exams) often do not
explicitly probe students’ conceptual understanding or
sense-making skills in a consistent manner.[10] And yet,
these are among the key learning goals identified by
faculty for upper-division QM. Therefore, as a major
component of this effort, we have created a conceptuallyfocused assessment tool which is described below.
After the results were analyzed, we revised several of
the questions and replaced others. Thirteen more student
interviews were conducted in order to assess question validity (eight interviews) and to learn more about student
thinking (five interviews). Finally, faculty were consulted
both in a working group and via an email survey after
which a final set of 14 questions was selected.
This new QMAT version[8] was administered in a
similar fashion near the end of the Spring 2009 semester
to 36 students (46 were registered for the course). The
class was taught by a non-PER faculty member using the
materials and reforms mentioned above.
Goals of the QMAT
Results and Discussion
The assessment tool was developed with three goals
in mind: reflect faculty learning goals, provide an assessment of student learning difficulties, and act as a tool to
help guide faculty efforts at improving QM instruction.
Reflect Course Learning Goals: For the QMAT to
be useful to and valued by faculty, its questions need
to measure student learning of content and skills which
most faculty view as important and feel their students
should master.
Assess student learning difficulties: Several researchers have found persistent learning difficulties experienced by students undertaking the formal study of
quantum mechanics.[2, 3, 11–15] Therefore, the QMAT
should measure the extent to which students are still experiencing these difficulties after the course.
Assist faculty in course improvement: Physics
faculty are currently engaged in efforts to reform QM
instruction at many colleges and universities. By comparing data across semesters and at different institutions,
faculty should be able to use the QMAT as a tool to aid
their course-improvement efforts.
The QMAT contains questions addressing several areas of the quantum mechanics curriculum. Figure 1
shows student performance in five categories of learning
goals which are probed by multiple questions. During
our analysis of the exams, student learning difficulties
were particularly apparent in the areas of measurement
and time evolution. These difficulties, along with sample
questions, are discussed below.
Development Process
The questions on the QMAT were developed based on
faculty input, adapted from published assessment questions, or inspired by observed student difficulties.[3, 16,
17]. Students who had previously completed the course
were interviewed as they worked through these questions. Revisions were made and a final set of sixteen
questions were selected after a total of thirteen student
interviews. This preliminary version was administered
during the last week of class in the Fall 2008 semester.
Students were given one fifty-minute class period to
complete the QMAT. While the assessment was optional,
students were encouraged to try their best and were given
feedback on their performance in different subject areas
(e.g., probability density, time development). Twentyseven (out of a total of thirty-three) students took the
QMAT.
100
80
Average %
QUANTUM MECHANICS ASSESSMENT
TOOL (QMAT)
60
40
20
Meas. TISE
WF
Time Prob.
FIGURE 1. QMAT scores by learning-goal category for
the Spring 2009 semester. The categories are: measurement
(Meas.), the time independent Schrödinger equation (TISE),
wave functions / boundary conditions (WF), time evolution
(Time), and probability / probability density (Prob.). Error bars
indicate the standard error of the mean.
Student difficulties with measurement
Several researchers have noted student difficulties working with measurement in quantum mechanics.[3, 14, 18–
20] The QMAT is designed to probe student learning in
this area with parts of three different questions.
Parts (a) and (b) of the question shown in Figure 2
measured significant confusion about the state of a particle in an infinite square well after a series of incompatible measurements. 72% of students correctly noted that
the wave function after the first measurement is u2 (x),
however, only 31% recognized that all allowed energies
were possible after a subsequent position measurement
(35% of the students who correctly answered part (a)). A
Consider a particle in a 1D, infinite square well with
width a, centered at a/2. The normalized energy
eigenstate wave functions are un (x) with energies En
(n = 1 is the ground state).
The particle starts in a state given by
!
r
r
4
1
Ψ(x,t = 0) =
u1 (x) +
u2 (x) .
5
5
a) You make an energy measurement on this system and find the maximum possible value for
the energy. What is the state, ψ(x), of the
system after this measurement?
(72% correct)
b) After the energy measurement, you make a position measurement. After this position measurement, you immediately re-measure the
energy. At this point, what value(s) could
you get for energy?
(31% correct)
c) Does your answer to part b depend on how
long you wait between the position and energy measurements? Explain.
(11% correct)
FIGURE 2. Measurement problem from QMAT. This question builds on work from ref. [3] (p. 409) and from ref. [15]
common error in part (a) was the choice of the lower energy state because it had a higher probability. The common errors on part (b) — from students who said that
the choices were E1 or E2 (42% of all answers) and from
students who said or implied that the position measurement would not alter the energy (19%) — reveal that students are not fully grasping the important relationships
between energy, a particle’s wave function, and measurement.
Students responses to statement I in Figure 3 demonstrate that many students fail to properly grasp the distinction between the Hamiltonian operator and physical measurement — only 28% were able to explain the
difference satisfactorily. Two common answers among
the 61% of students who incorrectly agreed that acting
on a quantum state with the Hamiltonian is mathematically equivalent to measuring the energy were: 1) because Ĥ |ψi = E |ψi (32%), and 2) a statement to the
effect that any quantum state has a well-defined energy
which is what is measured when the Hamiltonian is acted
on that state (23%).
Student difficulties with time development
Another area where researchers have documented student difficulties is in the area of time development.[3,
Which of the statements below are true for any quantum state, |ψi (Circle either T (the statement is always true) or F for each statement)?
I. T or F: Acting on |ψi with the Hamiltonian is the
mathematical equivalent of making a
measurement of the energy of that state.
(28% correct)
II. T or F: Applying the Hamiltonian to |ψi gives you
information about how that state will evolve in
time.
(25% correct)
FIGURE 3. True / False questions from QMAT. Statement I
was adapted from ref. [17]
14, 20] The QMAT looked at student learning in this area
with 2 problems and in parts of 2 others.
Part (c) of the problem in Figure 2 exposed student difficulties in working with the time development of quantum superposition states by asking students if waiting before making an energy measurement can affect the possible measured energies. Only 4 students (11%) managed
to convincingly describe why the amount of time before
the second energy measurement did not matter (only 3
of whom earned full credit on the rest of the problem).
Among students who felt that the time delay would matter, the most common reasons were that the energy eigenstates would evolve at different speeds or that the delta
function resulting from the position measurement would
spread out which would change the energy of the state.
A different group of 4 students who felt that the position
measurement did not change the fact that the system was
in an energy eigenstate (part (b)) thought that time did
not matter because it was a stationary state. In the words
of one student: “No. Measuring E sets the wave eq with
that val forever. Collapses it to that un .” Students who
felt that time does not matter are not making use of the
unique role played by energy in QM as shown by this
representative answer; “No. regardless of how long you
wait the wavefunction will have collapsed to an eigenvector of position space.”
Statement II in Figure 3 demonstrated that students
have difficulties with the Hamiltonian operator’s role in
the time development of quantum states. Only 44% correctly agreed that the Hamiltonian operator is associated with time development and only 25% wrote a convincing explanation (mostly appealing to the full, timedependent Schrödinger equation or breaking the wave
function into energy eigenstates and multiplying each
term by the appropriate time-dependence term). The students who disagreed with the statement seem to draw
no connection between the Hamiltonian and the full
Schrödinger equation saying things like: “H |ψi tells you
nothing about time”, “The Hamiltonian gives information about energy only”, “when we use Ĥ, it’s to solve
the TISE, so Ĥ doesn’t really tell us about time.”, and
p̂2
+V this has no time dependence.”
“Ĥ = 2m
Student concepts on time development of quantum
states are further probed by the question in Figure 4.
Is the following statement true for all operators, Q̂?
Explain briefly why you agree or disagree.
A system which is in an eigenstate of Q̂ will stay
in that state until disturbed by measurement. (36%
correct)
FIGURE 4. QMAT question about time development of an
eigenstate
Only 36% of students gave a convincing explanation
of why the statement was not true for an arbitrary operator (usually either a counter example or a statement that it
is only true for energy eigenstates[21]). Among students
who agreed with the statement (43%), the two most popular reasons were that it was a postulate of QM or that it
was true simply because it was an eigenstate. A representative quote from the latter category is: “Yes, Eigenstate
= stationary state”. Meanwhile 15% of the students who
disagreed with the statement indicated that nothing could
be said until a measurement of Q̂ was made. For instance,
one student wrote “Until measurement is made, we don’t
know what state the system is in. (accompanied by a picture of a cat in a box)”.
CONCLUSIONS
These results demonstrate that the QMAT functions as an
assessment tool by exposing common student difficulties
in areas such as measurement and time development
in QM. It also demonstrates that students in CU’s QM
I course are not achieving all of our learning goals,
despite reforms that include clicker questions and other
interactive techniques targeting these ideas. Thus, the
QMAT can serve to raise faculty awareness of student
difficulties and guide future reform efforts.
Specifically, we find that across the QMAT, students
frequently respond as though: all quantum states (including superposition states) have a definite energy, and
time dependence only requires ‘tacking on’ a single
term exp[−iEt/h̄] to any quantum state (including superposition states). These observations are consistent
with existing QM research literature.[2, 3, 14, 15] Our
preliminary interviews suggest that students are overgeneralizing from the time independent Schrödinger
equation, Ĥ |ψi = E |ψi. Students also frequently respond as though sequential measurements on a quantum
state retain all original information encoded in the starting state, again consistent with literature.[3]
Moving forward, the QMAT will continue to be used
at CU and is available to faculty interested in administering the test to their QM classes. These results, when
pooled, will aid the community in the search to find effective new approaches to teaching upper-division QM.
ACKNOWLEDGMENTS
We thank Carl Wieman and Sam McKagan for useful
discussions. We also thank the physics faculty members
who generously contributed to the creation of both the
learning goals and the QMAT: A. Becker, T. DeGrand,
O. DeWolfe, N. Finkelstein, C. Greene, A. Hasenfratz,
E. Kinney, K. Mahanthappa, U. Nauenberg, J. Price,
C. Rogers, K. Stenson, and E. Zimmerman. This work
is funded by The CU Science Education Initiative and
NSF-CCLI Grant # 0737118.
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While this is only true for some QM systems, students
were given credit for this type of answer.